What is the value of ∫ from 0 to 1 of (x^2 + 1) dx? Step by Step

Step by step

The question

Question: find the integral from 0 to 1 of x² + 1.

∫ from 0 to 1 of (x² + 1) dx

An integral from a to b gives one number.

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The antiderivative

First the antiderivative: add 1 to each power, divide by the new power.

Now we have F(x) = x³ over 3 + x.

F(x) = x³3 + x

No + C here: in F(b) minus F(a), the two Cs cancel.

no + C here: it cancels

2 Top minus bottom

The rule: put in the top number, then take away the bottom number.

∫ from a to b of f(x) dx = F(b) − F(a)

Write F(x) with 0 and 1 on the bar.

= (x³3 + x) from 0 to 1

Put in 1 first, then take away the same with 0.

= (1³3 + 1) − (0³3 + 0)

First the powers and the times.

= (13 + 1) − (0 + 0)

Then each bracket: the first is 4/3, the second is 0.

= 43 − 0

Careful!

Top first, then take away the bottom.

The other way gives the wrong sign.

Careful: not F(0) − F(1) ✗

So the integral is 4/3.

= 43

3 Check

Let's check F: its derivative is x² + 1, the function we started with.

Right!

F′(x) = x² + 1

4 Pick the choice

Our answer is 4/3.

That is choice A.

A) 43

5 Why not the others

B is 1/3.

Wrong: lost the + 1.

B) 13 ✗ lost the + 1

C is 1.

Wrong: put in x = 0 only.

C) 1 ✗ put in x = 0 only

D is 2.

Wrong: put 1 into x² + 1: put it into the antiderivative.

D) 2 ✗ put 1 into x^2 + 1: put it into the antiderivative

Let's look back.

First: the antiderivative.

Then: top minus bottom.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer